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V. AISC ASD - 2D Frame Validation

Determine the allowable stresses (per 1989 AISC code) for the members of the structure as shown in the figure. Also, perform a code check for these members based on the results of the analysis.

Reference

Attached step by step hand calculation as per 1989 AISC ASD code. Ninth Edition.

Problem

Steel braced frame

Members 1, 2 = W12X26, Members 3, 4 = W14X43

Members 5, 6, 7 = W16X36, Memb 8= L40404, Memb 9 = L50506

F = 15 kip (wind load)

w = 2 kip/ft (dead load + live load)

Steel: Fy = 36ksi

Hand Calculation

Manual / Code refers to AISC Manual of Steel Construction, Allowable Stress Design, ninth edition.

Member 1

Size W12x26, L = 10 ft., a = 7.65 in2, Sz = 33.39 in3

Per Section F2 is the lower value for the yielding and lateral-torsional buckling limit states:

Mn = Mp = FyZx = 36ksi × 37.2 in3 = 1,339 in·k = 111.3 ft·k

 

L p = 1.76 r y E F y = 1.76 ( 1.51 in ) 29 , 000 ksi 36 ksi = 75.4 in

 

From clause F1-2, page 5-45 of Manual, Lc = 6.85 ft.

From observation Load case 1 will govern,

Fx = 25.0 kip (compression), Mz = 56.5 ft·kip

Area of compressive flange = 6.49*0.38 = 2.466 in2

Allowable bending stress, per Clause F1-8 (page 5-47 of Manual)

fb = 12. · 1000 · 1.0/[10 · 12 · (12.22/2.466)] = 20.1817 ksi

(kl/r)y = 120/1.5038 = 79.8, so fa = 15.38 ksi

Stresses per Table C-36 (page 3-16 of manual)

fa = 25./7.65 = 3.268

fb = 56.5 · 12/33.39 = 20.31 ksi

(kl/r)z = 120/5.1639 = 23.238, so F'ez = = 276.54 ksi

Equation H1-1, page 5-54 of Manual

3.268/15.38 + 0.85 · 20.31 / [(1-3.268/276.54) · 20.1817]  =  1.078

Equation H1-2, page 5-54 of Manual.

3.268/(0.6 · 36) +20.31/20.1817 = 1.1576

Therefore, equation H1-2 governs and ratio = 1.1576

Member 2

Size W12x26, L = 5 ft., a = 7.65 in2, Sz = 33.39 in3

From observation load case 1 will govern,

Fx = 8.71 kip (compression), Mz = 56.50 ft·kip

Since L is less than Lc = 6.85ft, per Clauses F1-1 & F1-2 (page 5-45 of Manual)

fb = 0.66 · 36 = 23.76 ksi

(kl/r)y = 60/1.5038 = 39.90, so fa = 19.19 ksi

Per Table C-36 (page 3-16 of Manual)

fa = 8.71/7.65 = 1.1385, fb = 56.5 · 12/33.39 = 20.31 ksi

Since fa/Fa less than 0.15, apply equation H1-3 (page 5-54 of Manual)

1.1385/19.19 + 20.31/23.76 = .0593 + .8548 = 0.9141

Member 3

Size W14X43, L = 11ft., a = 12.6 in2, Sz = 62.7 in3

From observation load case 3 will govern,

Fx = 25.5 kip (compression), Mz = 112.173 ft·kip

Referring to clause F1-2, page 5-45 of Manual.

Lc = 8.4 ft. Therefore

fb = 0.6 · 36 = 21.6 ksi

(kl/r)y = 132/1.8941 = 69.69,

so fa = 16.46 ksi per Table C-36 (page 3-16 of Manual)

fa = 25.5/12.6 = 2.024

fb = 112.173 · 12/62.66 = 21.48 ksi

since fa/Fa less than 0.15, use equation H1-3, page 5-54 of Manual

2.024/16.46 + 21.48/21.6 = 0.123 + 0.994 = 1.117

Member 4

Size W14x43, L = 4ft, a = 12.6 in2, Sz = 62.7 in3

From observation, load case 3 will govern,

Fx = 8.75 kip (tension), Mz = 112.173 ft·kip

Since L is less than Lc = 8.4 ft

fb = 0.66 · 36 = 23.76 ksi

Clause F1-1 (page 5-45 of Manual)

fa = 8.75/12.6 = 0.694

fb = 112.73x12/62.66 = 21.48 ksi

Combined tension and bending, use equation H2-1, page 5-55 of Manual.

0.694/(0.6 · 36) + 21.48/23.76 = 0.032 + 0.904 = 0.936

Member 5

Size W16x36, L = 5ft, a = 10.6 in2, Sz = 56.49 in3

From observation, load case 3 will govern.

Fx = 14.02 kip (compression), Mz =57.04 ft·kip

Since L is less than Lc = 7.37 (per Clause F1-2, page 5-45 of Manual)

fb = 0.66 · 36 = 23.76 ksi

(kl/r)y = 60./1.52 = 39.47

so fa = 19.23 ksi per Table C-36 (page 3-16 of Manual)

fa = 14.02/10.6 = 1.32

fb = 57.04 · 12/56.5 = 12.12 ksi

Since fa/Fa less than 0.15, use equation H1-3, page 5-54 of Manual

1.32/19.23 + 12.12/23.76 = 0.069 + 0.510 = 0.579

Member 6

Size W16x36, L = 16ft, a = 10.6 in2, Sz = 56.49 in3

From observation, load case 1 will govern. Forces at midspan are

Fx = 5.65 kip (compression), Mz = 71.25 ft·kip

From Chapter F of the AISC ASD 9Pth P ed. specs., with Cb = 1.0,

(102,000Cb/Fy)1/2 =53.229

(510,000Cb/Fy)1/2 =119.02

L/rT = 192/1.79 = 107.26

53.229 < 107.26 < 119.02

Therefore Fb (as per F1-6, page 5-47 of Manual)

[(2/3) – 36 · 107.26 · 107.26/(1,530,000)] · 36 = 14.25 ksi

(Kl/r)y = 192/1.5203 = 126.29

so fa = 9.36 per Table C-36 (page 3-16 of Manual)

fa = 5.65/10.6 = 0.533

fb = 71.25x12/56.49 = 15.14 ksi

Since fa/Fa less than 0.15 use formula [H1-3, page 5-54 of Manual]

0.533/9.36 + 15.14/14.25 = 0.057 + 1.062 = 1.119

Member 7

Size W16x36, L =4ft, a = 10.6in2, Sz =56.49in3

Lc = 7.37ft (Clause F1-2 page 5-45 of Manual)

From observation load case 3 will govern, Fx = 24.06 kip (tension), Mz = 62.96 ft·kip

From Clause F1-1, the allowable compressive stress is

fb = 0.66 Fy = 23.76 ksi

Since section is in tension, per Clause F1-5 (page5-45 of Manual)

fb = 0.60 · 36 = 21.60 Ksi

Choosing the larger of above 2 values, fb = 23.76 Ksi

fa = 24.06/10.6 = 2.2698

fb = 62.96 · 12/56.49 = 13.37

Since combined tension and bending, use equation H 2-1, page 5-55 of the AISC ASD 9Pth P ed. specs.

2.2698/(0.6 · 36) + 13.37(23.76) = 0.105 + 0.5627 = 0.6677

Member 8

Size L4x4x1/4, L = 7.071 ft, a = 1.94 in2

From observation load case 1 will govern, Fx = 23.04 kip (Comp.)

Fa is computed as per page 5-310 of the AISC ASD 9Pth P ed.specs.

Qs = 1.34 – 0.00447 · (4/0.25)·(36)1/2 = 0.9108

Qa = 1.0

Q = Qs · Qa = 0.9108

Cc= 92.0·π2·E/(Q·Fy)]1/2 = [2.0·π2 · 29000/(0.9108 · 36)]1/2 = 132.1241

Kl/r = 7.071 · 12/0.795 = 106.73 < Cc.

Hence, fa = 11.6027 ksi (computed per equation 4-1)

Actual compressive stress

fa = 23.04/1.94 = 11.876 ksi

Therefore, Ratio =

fa/fa = 11.876/11.602 = 1.024

Member 9

Size L5x5x3/8, L = 5.657 ft, a = 3.61 in2

From observation, load case 1 governs, Fx = 48.44 kip (Comp.)

Fa is computed as per page 5-310 of the AISC ASD 9Pth P ed. specs.

Qs = 1.34 – 0.00447 · (5/0.375)·(36)1/2 = 0.9824

Qa = 1.0

Q = Qs · Qa = 0.9824

Cc= 92.0·π2·E/(Q·Fy)]1/2 = [2.0·π2 · 29000/(0.9824 · 36)]1/2 = 127.2238

(Kl/r)min = 5.657 · 12/0.99 = 68.57 < Cc.

Hence, fa = 16.301 ksi (computed per equation 4-1)

Actual compressive stress

fa = 48.44/3.61 = 13.418 ksi

Therefore Ratio =

fa/fa = 13.418/16.301 = 0.823

Comparison

Governing stress ratio at member no. Hand Calculation STAAD.Pro Difference
1 1.158 1.154 negligible
2 0.914 0.913 negligible
3 1.117 1.121 negligible
4 0.936 0.939 negligible
5 0.579 0.580 negligible
6 1.119 1.107 1.1%
7 0.668 0.670 negligible
8 1.024 1.045 2.1%
9 0.823 0.815 1.0%

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISC\ASD 9th Edition\AISC ASD - 2D Frame validation.STD is typically installed with the program.

STAAD PLANE VERIFICATION PROBLEM NO 13
START JOB INFORMATION
ENGINEER DATE 22-Sep-18
END JOB INFORMATION
*
*  THIS DESIGN EXAMPLE IS VERIFIED BY HAND CALCULATION
*  FOLLOWING AISC-89 CODE.
*
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 25 0 0; 3 0 10 0; 4 25 11 0; 5 0 15 0; 6 25 15 0; 7 5 15 0;
8 21 15 0;
MEMBER INCIDENCES
1 1 3; 2 3 5; 3 2 4; 4 4 6; 5 5 7; 6 7 8; 7 8 6; 8 3 7; 9 4 8;
MEMBER PROPERTY AMERICAN
1 2 TABLE ST W12X26
3 4 TABLE ST W14X43
5 TO 7 TABLE ST W16X36
8 TABLE ST L40404
9 TABLE ST L50506
MEMBER TRUSS 
8 9
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 4.176e+06
POISSON 0.3
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 2 PINNED
LOAD 1 DL + LL
MEMBER LOAD
5 TO 7 UNI Y -2
LOAD 2 WIND FROM LEFT
JOINT LOAD
5 FX 15
LOAD COMBINATION 3
1 0.75 2 0.75 
PERFORM ANALYSIS
LOAD LIST 1 3
PRINT MEMBER FORCES
PARAMETER 1
CODE AISC
TRACK 1 ALL
CHECK CODE ALL
FINISH

STAAD Output

                         STAAD.Pro CODE CHECKING - (AISC 9TH EDITION)   v1.0
                         ***********************
 ALL UNITS ARE - KIP  FEET (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================
 *    1  ST   W12X26                   (AISC SECTIONS)
                           FAIL     AISC- H1-2         1.154         1
                       25.00 C          0.00          56.50       10.00
  -----------------------------------------------------------------------
 | MEM=     1, UNIT KIP-INCH, L= 120.0 AX=   7.65 SZ=    33.4 SY=     5.3|
 | KL/R-Y=  79.8  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 20.21 |
 | FTZ= 21.60  FCY= 27.00  FTY= 27.00  FC= 15.38  FT= 21.60  FV= 14.40   |
  -----------------------------------------------------------------------
      2  ST   W12X26                   (AISC SECTIONS)
                           PASS     AISC- H1-3         0.913         1
                        8.72 C          0.00          56.50        0.00
  -----------------------------------------------------------------------
 | MEM=     2, UNIT KIP-INCH, L=  60.0 AX=   7.65 SZ=    33.4 SY=     5.3|
 | KL/R-Y=  39.9  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 23.76 |
 | FTZ= 23.76  FCY= 27.00  FTY= 27.00  FC= 19.20  FT= 21.60  FV= 14.40   |
  -----------------------------------------------------------------------
 *    3  ST   W14X43                   (AISC SECTIONS)
                           FAIL     AISC- H1-3         1.121         3
                       25.50 C          0.00        -112.20       11.00
  -----------------------------------------------------------------------
 | MEM=     3, UNIT KIP-INCH, L= 132.0 AX=  12.60 SZ=    62.5 SY=    11.3|
 | KL/R-Y=  69.7  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 21.60 |
 | FTZ= 21.60  FCY= 27.00  FTY= 27.00  FC= 16.46  FT= 21.60  FV= 14.40   |
  -----------------------------------------------------------------------
      4  ST   W14X43                   (AISC SECTIONS)
                           PASS     AISC- H2-1         0.939         3
                        8.83 T          0.00        -112.20        0.00
  -----------------------------------------------------------------------
 | MEM=     4, UNIT KIP-INCH, L=  48.0 AX=  12.60 SZ=    62.5 SY=    11.3|
 | KL/R-Y=  25.3  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 23.76 |
 | FTZ= 23.76  FCY= 27.00  FTY= 27.00  FC= 19.85  FT= 21.60  FV= 14.40   |
  -----------------------------------------------------------------------
      5  ST   W16X36                   (AISC SECTIONS)
                           PASS     AISC- H1-3         0.580         3
                       14.02 C          0.00         -57.00        5.00
      VERIFICATION PROBLEM NO 13                               -- PAGE NO.    6
 ALL UNITS ARE - KIP  FEET (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================
  -----------------------------------------------------------------------
 | MEM=     5, UNIT KIP-INCH, L=  60.0 AX=  10.60 SZ=    56.4 SY=     7.0|
 | KL/R-Y=  39.5  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 23.76 |
 | FTZ= 23.76  FCY= 27.00  FTY= 27.00  FC= 19.23  FT= 21.60  FV= 14.40   |
  -----------------------------------------------------------------------
 *    6  ST   W16X36                   (AISC SECTIONS)
                           FAIL     AISC- H1-3         1.107         1
                        5.65 C          0.00         -71.25        8.00
  -----------------------------------------------------------------------
 | MEM=     6, UNIT KIP-INCH, L= 192.0 AX=  10.60 SZ=    56.4 SY=     7.0|
 | KL/R-Y= 126.3  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 14.45 |
 | FTZ= 21.60  FCY= 27.00  FTY= 27.00  FC=  9.36  FT= 21.60  FV= 14.40   |
  -----------------------------------------------------------------------
      7  ST   W16X36                   (AISC SECTIONS)
                           PASS     AISC- H2-1         0.670         3
                       24.13 T          0.00          63.00        0.00
  -----------------------------------------------------------------------
 | MEM=     7, UNIT KIP-INCH, L=  48.0 AX=  10.60 SZ=    56.4 SY=     7.0|
 | KL/R-Y=  31.6  CB=  1.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ= 23.76 |
 | FTZ= 23.76  FCY= 27.00  FTY= 27.00  FC= 18.88  FT= 21.60  FV= 14.40   |
  -----------------------------------------------------------------------
 *    8  ST   L40404                   (AISC SECTIONS)
                           FAIL     AISC- H1-1         1.045         1
                       23.03 C          0.00           0.00        0.00
  -----------------------------------------------------------------------
 | MEM=     8, UNIT KIP-INCH, L=  84.9 AX=   1.93 SZ=     0.8 SY=     1.7|
 | KL/R- = 108.4  CB=  0.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ=  0.00 |
 | FTZ=  0.00  FCY=  0.00  FTY=  0.00  FC= 11.42  FT= 21.60  FV=  0.00   |
  -----------------------------------------------------------------------
      9  ST   L50506                   (AISC SECTIONS)
                           PASS     AISC- H1-1         0.815         3
                       48.55 C          0.00           0.00        0.00
      VERIFICATION PROBLEM NO 13                               -- PAGE NO.    7
 ALL UNITS ARE - KIP  FEET (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================
  -----------------------------------------------------------------------
 | MEM=     9, UNIT KIP-INCH, L=  67.9 AX=   3.65 SZ=     1.8 SY=     3.9|
 | KL/R- =  68.8  CB=  0.00  YLD= 36.00  ALLOWABLE STRESSES:  FCZ=  0.00 |
 | FTZ=  0.00  FCY=  0.00  FTY=  0.00  FC= 16.32  FT= 21.60  FV=  0.00   |
  -----------------------------------------------------------------------